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Engines of Liberation
Jeremy Greenwood, Ananth Seshadri and Mehmet Yorukoglu1, 2
November 2000, PreliminaryComments Welcome
1Affiliations : University of Rochester, University of Wisconsin and The University ofChicago.
2Updates: http://www.econ.rochester.edu/Faculty/GreenwoodPapers/engine.pdf
Abstract
Electricity was born at the dawn of the last century. Households were inundatedwith a flood of new consumer durable goods. What was the impact of this consumerdurable goods revolution? It is argued here that the consumer goods revolutionliberated women from the home. To analyze this hypothesis, a Beckerian model ofhousehold production is developed. Households must decide whether to adopt thenew technologies or not, and whether a married woman should work. Can such amodel explain the rise in married female labor-force participation that occurred inthe last century? Yes.
Keywords: The second industrial revolution, technology adoption, householdproduction theory, female labor-force participation.
Subject Area: Macroeconomics.
JEL Classification Numbers: E1, J2, N1.
“The housewife of the future will neither be a slave to servants or
herself a drudge. She will give less attention to the home, because the
home will need less; she will be rather a domestic engineer than a domestic
laborer, with the greatest of all handmaidens, electricity, at her service.
This and other mechanical forces will so revolutionize the woman’s world
that a large portion of the aggregate of woman’s energy will be conserved
for use in broader, more constructive fields.”
Thomas Alva Edison, as interviewed in Good Housekeeping Magazine,
LV, no. 4 (October 1912, p. 436)
1 Introduction
1.1 Facts
The dawning of the last century ushered in the second industrial revolution: the rise
of electricity, the internal combustion engine, and the petrochemical industry. As
this was happening, another technological revolution was beginning to percolate in
the home: the housework revolution. The impact of the housework revolution was
no less than the industrial revolution.
Durable Goods: The housework revolution was spawned by massive investment-
specific technological progress in the production of household capital. This era saw
the rise of central heating, dryers, electric irons, frozen foods, refrigerators, sewing
machines, washing machines, vacuum cleaners, and other appliances now considered
fixtures of everyday life. The spread of electricity, central heating, flush toilets and
running water, through the U.S. economy is shown in Figure 1.1 Likewise, Fig-
ure 2 plots the diffusion of some common electrical appliances through American
1The data for this figure is taken from Lebergott (1976, p. 100) and Lebergott (1993, Tables
II.14 and II.15).
1
1860 1880 1900 1920 1940 1960 1980 2000
Year
0
20
40
60
80
100
Hou
seho
ld A
dopt
ion,
%
Central Heating
Electricity
Running Water
Flush Toilet
Figure 1: Spread of Basic Facilities Through the U.S. Economy
households.2
Investment in household appliances as a percentage of GDP has almost doubled
over the last century. It represented about 0.5% of GDP in 1988, which was about
2.9% of total investment spending. Likewise, the stock of appliances as a percentage
of GDP has also risen continuously, as Figure 3 illustrates.3
To understand the impact of the housework revolution, try to imagine the tyranny
of household chores at the turn of the last century. In 1890 only 24 percent of houses
had running water, none had central heating. So, the average household lugged
around the home, 7 tons of coal and 9,000 gallons of water per year. The simple
2Sources: Survey of Current Business and Fixed Reproducible Tangible Wealth in the United
States, 1925-89. Washington, D.C., U.S. Department of Commerce.3Sources: (i) Dishwashers, refrigerators, and vacuum cleaners — Electrical Merchandising; (ii)
Dryers and microwaves — Burwell and Swezey (1990, Figures 11.8 and 11.10); (iii)Washers — Leber-
gott (1993, Table II.20)
2
1920 1930 1940 1950 1960 1970 1980 1990
Year
0
20
40
60
80
100
Hou
seho
ld A
dopt
ion,
%
refrigerator
vacuum
washer
microwave
dishwasher
dryer
Figure 2: Diffusion of Electrical Appliances in the Household Sector
task of laundry was a major operation in those days. While mechanical washing
machines were available as early as 1869, this invention really took off only with
the development of the electric motor. Ninety-eight percent of households used a 12
cent scrubboard to wash their clothes in 1900. Water had to be ported to the stove,
where it was heated by burning wood or coal. The clothes were then cleaned via a
washboard or mechanical washing machine. They had to be rinsed out after this.
The water then needed to be wrung out, either by hand or by using a mechanical
wringer. The clothes were dried using a clothes line. Then, the oppressive task of
ironing began, using heavy flatirons that had to be heated continuously on the stove.
The electric iron was first patented in 1882 by Henry W. Seely. Westinghouse
launched an advertising campaign to acquaint the public with the benefits of the iron
in 1906. The iron diffused quickly with the spread of electricity. Seventy-one percent
3
1900 1920 1940 1960 1980 2000Year
0.0
0.1
0.2
0.3
0.4
0.5
Appl
ianc
e-In
vest
men
t-to-
GD
P R
atio
, %
0.6
1.1
1.6
2.1
2.6
Appl
ianc
e-to
-GD
P R
atio
, %
Investment
Stock
Figure 3: Household Appliances
of wired homes had them in 1926.4 The first electric washing machine surfaced in
1908. It was invented by Alva J. Fisher and sold by the Hurley Machine Company.
As with many inventions, the initial incarnation of the idea was crude. Clothes were
spun around on a drum driven by an uncovered chain attached to an electric motor.
Maytag introduced an electric washing machine with an agitator in 1922. It was a
great success and by 1927 the company had produced a million of them. Thirty-
six percent of wired homes had an electric washing machine in 1926.5 The early
machines really just replaced the scrubboard. Homeowners still had to use a wringer.
The electric water heater arrived around this time, too. Fully automatic washing
machines with a spin-cycle didn’t appear until about 1940. The clothes dryer didn’t
catch on until the beginning of the 1950s.
4Source: Electrical Merchandising, January 1926, pp. 6002.5Source: Ibid.
4
Time Savings: The amount of time freed by modern appliances is somewhat spec-
ulative. Controlled engineering studies documenting the time saved on some specific
task by the use of a particular machine would be ideal. Unfortunately, these studies
seem hard to come by. The Rural Electrification Authority supervised one such study
based on 12 farm wives during 1945-46. They compared the time spent doing laundry
by hand to that spent using electrical equipment. The women also wore a pedometer.
One subject, Mrs. Verett, was reported on in detail.6 Without electrification, she
did the laundry in the manner described above.7 After electrification Mrs.Verett had
an electric washer, dryer and iron. A water system was also installed with a water
heater. They estimated that it took her about 4 hours to do a 38 lb load of laundry
by hand, and then about 4 and 1/2 hours to iron it using old-fashioned irons. By
comparison it took 41 minutes to do a load of the laundry using electrical appliances
and 1 and 3/4 hours to iron it. The woman walked 3,181 feet to do the laundry
by hand, and only 332 feet with electrical equipment. She walked 3,122 feet when
ironing the old way, and 333 the new way.
In 1900 the average household spent 58 hours a week on housework — meal prepa-
ration, laundry and cleaning — Lebergott (1993, Table 8.1) estimates. This compares
with just 18 in 1975. Sociologists suggest that modern appliances have had little
effect on the total time allocated to housework. This is based on time diary evidence.
They feel that with the mechanization of the household, societal standards for good
housekeeping have risen to keep women enslaved. Vanek (1973, Tables 14.4 and 14.5)
reports that total amount of time spent on housework in a family with an employed or
non-employed mother seems remarkably constant over the last 70 years or so, about
26 hours a week for the former and 55.4 hours for the later in 1965-1966. Even taking
this pessimistic attitude, this implies that the average amount of total time spent
on housework has fallen as female labor-force participation has risen. The implied
average amount of total time spent on housework (ignoring part-time work) is shown
6This study is reported in Electrical Mechandising, March 1, 1947: pp. 38-39.7She actually had a gas-powered washing machine that replaced the scrubboard.
5
1910 1930 1950 1970
20
30
40
50
60
Tota
l Hou
rs p
er W
eek
on H
ouse
wor
k
30
50
70
90
Dom
estic
Wor
kers
per
1,0
00 H
ouse
hold
s
Domestics
Housework -- Vanek, implied
Housework -- Lebergott
Figure 4: Housework
in Figure 4. At the same time the number of paid domestic workers declined, presum-
ably in part due to the labor-saving nature of household appliances.8 A reasonable
conclusion is that the time spent on the more onerous household chores, such as those
associated with cooking, cleaning, doing the laundry, etc., declined considerably in
the last century.
Labor-Force Participation: What was the effect of this massive technological ad-
vance in the production of household capital on labor force participation? A case
can be made that it liberated women from the home. As can be seen from Figure 5,
female labor-force participation rose steadily since 1890. At the same time the num-
ber of homemakers continuously declined. Real income per full-time female worker
grew five fold over this period. In 1890 a female worker earned about 50 percent of
a what a male did, and by 1970 this number had risen to only 60 percent. It seems
8Source: Oppenheimer (1969, Table 2.5)
6
1880 1900 1920 1940 1960 1980Year
0
20
40
60
Perc
enta
ge
1000
2000
3000
4000
5000
Earn
ings
, $19
67
Homemakers, left
Earnings Ratio, left
Participation, left
Female Earnings, right
Figure 5: Female Labor-Force Statistics
unlikely that the small rise in the relative income of a female worker could explain
the dramatic rise in labor force participation. It is more likely that the rise in overall
real wages, in conjunction with the introduction of labor-saving household appliances,
explains the rise in female labor-force participation.9
A Wrinkle: Now, the data suggests that women from poor households entered the
labor force first. As can be seen, the higher a woman’s husband’s income, the less
likely she was to work. At the same time, the poorer a family, the slower they were
to purchase durable goods. The relative price of new goods tends to fall rapidly after
their introduction. Perhaps poor women went to work in order to save up the funds
required to buy the new durable goods at some later date, when their prices would
fall.9Sources: (i) Female earnings, ratio of female to male earnings, and participation — Goldin (1990,
Table 5.1); (ii) Homemakers — Vanek (1973, Table 1.22).
7
Labor Force Participation by Husband’s Income: US, April 1940
No Children under 10 With Children under 10
White Nonwhite White Nonwhite
$1-199 30.4% 44.5 15.4 23.2
200-399 25.5 37.5 12.2 21.0
400-599 24.0 37.2 11.0 19.2
600-999 23.2 33.7 11.2 18.6
1,000-1,499 21.3 25.5 8.7 14.1
1,500-1,999 16.8 20.9 5.5 16.3
2,000 and up 11.1 20.3 3.1 12.1
Source: Durand (1948, Table 17, p. 92)
8
Durable Goods Ownership by Socio-Economic Class: US, 1965
Working Not Working
All Family Income All Family Income
$3,000 $5,000 $8,000 $3,000 $5,000 $8,000
- 4,999 - 7,999 and up - 4,999 - 7,999 and up
Running Water 100% 100 100 100 99 100 99 100
Hot Water 99 100 99 98 98 99 97 100
Flush Toilet 98 96 98 100 98 99 97 100
Bath or Shower 98 91 99 98 98 98 97 100
Furnace 88 74 86 98 87 81 93 93
Telephone 90 65 94 94 92 88 99 100
Refrigerator 100 100 100 100 100 100 100 100
Freezer 14 4 13 19 12 8 17 14
Electric or Gas Stove 100 100 100 100 98 98 99 100
Washing Machine 90 78 92 92 98 97 99 100
— Nonautomatic 52 61 60 31 63 73 55 29
— Automatic 40 17 33 63 39 26 51 79
Dryer 60 43 56 75 57 45 70 86
Iron 100 100 100 100 100 100 100 100
Vacuum Cleaner 87 78 88 90 89 80 97 100
Sewing Machine 63 48 67 60 68 58 75 100
Source: Vanek (1973, Table 4.21, p. 155)
9
Durable Goods Ownership by Income: Canada, 1957
$2,000-$2,999 $3,000-$3,999 $4,000-$4,999 $5,000-$5,999 $6,000-$7,999
Refrigerator 79% 93 93 95 96
Stove 57 76 82 88 92
Washer 85 90 90 90 92
Television 83 88 90 91 91
Freezer 7 4 5 8 5
Vacuum Cleaner 40 61 67 80 83
Floor Polisher 21 29 42 51 51
Sewing Machine 53 64 61 65 62
Radio 88 91 92 93 95
Source: Day (1992, Table 8, p. 319)
1.2 The Analysis
To address the question at hand, Becker’s (1965) classic notion of household produc-
tion is introduced into a dynamic general equilibrium model. In particular, household
capital and labor can be combined to produce home goods, which yield utility. This
isn’t the first time that household production theory has been embedded into the
neoclassical growth model. Benhabib, Rogerson and Wright (1991) have done so to
study the implication of the household sector for business cycle fluctuations.10 Par-
ente, Rogerson andWright (2000) apply a similar framework to see whether household
production can explain cross-country income differentials. Home production has also
been introduced into an overlapping generations model by Rios-Rull (1993) to exam-
ine its impact on the time allocations of skilled versus unskilled labor.
The analysis undertaken here differs significantly, though, from the above work.
It assumes that over the last century there has been tremendous investment-specific
technological progress in the production of household capital. These new and im-
10This line of research has been recently been extended by Gomme, Kydland, and Rupert (2000).
10
proved capital goods allow household production to be undertaken using less labor.
The formalization of the labor shedding nature of the new technologies is reminis-
cent of Goldin and Katz’s (1998) description of the effect that continuous-process
and batch methods had on labor demand during the second industrial revolution. It
also resembles Krusell et al ’s (2000) analysis of the impact that biased technological
progress had on the postwar skill premium. At the heart of the developed frame-
work are two interrelated decisions facing each household. First, they must choose
whether or not to adopt the new technology at the going price. Second, they must de-
cide whether the woman in the family should work in the market or not. The upshot
of the analysis is that household production theory is a powerful tool for explaining
the rise in U.S. female labor-force participation over the last century.
2 The Economic Environment
The world is made up of overlapping generations. Each generation lives n periods.
Hence, in any period there are n generations around.
Tastes: Let tastes for an age-j household be given by
nXi=j
βi−j[µ lnmi + ν lnni + (1− µ− ν) ln li],
where mi and ni are the consumptions of market and non-market produced goods
and li is the household’s leisure.
Income: A household is made up of a male and a female. They are endowed
with two units of time, which they split up between market work, home work, and
leisure. Work in the market is indivisible. Set market time at ω. It will be assumed
that males always work in the market. The household can choose whether or not
the female will work in the market. Each household is indexed by an ability level, λ,
shared by both members. This is drawn at the beginning of their life. They make all
decisions knowing the value of λ. Let ability λ be drawn from a lognormal distribution.
11
Normalize the mean of λ at unity. Therefore, assume that lnλ ∼ N(−σ2/2, σ2).
Denote the distribution function for ability by L(λ). The market wage for an efficiency
unit of male labor is given by w. A woman earns the fraction φ of what a man does.
Hence, in a given period, a family of efficiency level λ will earn the amount wλω if
the female stays at home and the amount wλω+wφλω if she works. The family may
also have assets. Denote these by a and let the gross interest rate be r.
Household Production: Home goods are produced according to the following Leon-
tief production function. Specifically,
n = mind, ζh,
where d represents the stock of household durables and h proxies for the time spent
on housework. The variable ζ represents labor-augmenting technological progress in
the household sector. Durables goods are assumed to be lumpy. All housework is
done by women.
The Durable Goods Revolution, A Preview : A household technology is defined
by the triplet (d, h, ζ). Recall that household capital, d, is lumpy, and assume that
housework, h, is indivisible. Let the time price of the technology be q — this is set
in terms of hours of work (at the mean skill level). The cost of the technology is
equal to price of the durable goods, d, needed to operated it. Before the arrival of
electricity suppose that d = δ, h = ρη, and ζ = δ/(ρη), where 0 < ρη < 1 − ω andρ > 1. Using this old technology, n = mind, ζh = δ units of non-market goods canbe produced. The price of the old household technology will be set to zero. Now,
imagine that electricity comes along together with a new set of durable goods. Define
this new technology by the triplet (d0, h0, ζ 0). Here d0 = κδ, h0 = η, and ζ 0 = κδ/η,
where κ > 1. Note that ζ 0 = κρζ, so that technological progress can be broken down
into the additional amount of capital services provided and the amount of household
labor freed up. That is, with the new technology capital services rise by a factor
of κ. The old technology requires more labor, a factor ρ more. The new technology
produces n0 = mind0, ζ 0h0 = κδ > δ units of non-market goods. Should a household
12
adopt the new technology? This will depend on its price, q0, of course.
Market Production: Market production is undertaken according to the standard
neoclassical production function
y = kα(zl)1−α, (1)
where y is output, k represents the aggregate market capital stock, and l is the
aggregate labor input. Labor-augmenting technological progress is captured by the
variable z. Market output can be used for the consumption of market goods, m,
gross investment in business capital, i, and for gross investment in household capital,
d. Hence
m+ i+ d = y. (2)
The law of motion for business capital is
k0 = χk+ i, (3)
where χ factors in physical depreciation.
3 The Household’s Decision Problem
Asset Accumulation and Labor-Force Participation Decisions: Consider the dynamic
programming problem facing an age-j household. Suppose that the household has
already made its decision about whether or not to adopt the new technology for the
current period. Then the household’s state of the world is summarized by the triplet
(a, τ ,λ). Here τ ∈ 0, 1, 2 is an indicator function giving the state of the householdtechnology. When τ = 0 the household does not use the new technology in the cur-
rent period. When τ = 1 the household purchases and uses the new technology in the
current period. Last, τ = 2 denotes the case where the household has adopted previ-
ously. The lifetime utility for an age-j household with assets, a, state of technology
13
τ , and ability level λ is represented by V j(a, τ ,λ). It is easy to see that the decisions
regarding female labor-force participation and asset accumulation are summarized by
V j(a, 0,λ) = maxmaxa0µ ln(wλω + φwλω + ra− a0) + ν ln(δ) P(1)
+(1− µ− ν) ln(2− 2ω − ρη) + βmax[V j+1(a0, 0,λ), V j+1(a0, 1,λ)],maxa0µ ln(wλω + ra− a0) + ν ln(δ)
+(1− µ− ν) ln(2− ω − ρη) + βmax[V j+1(a0, 0,λ), V j+1(a0, 1,λ)],
V j(a, 1,λ) = maxmaxa0µ ln(wλω + φwλω + ra− a0 − wq) + ν ln(κδ)
+(1− µ− ν) ln(2− 2ω − η) + βV j+1(a0, 2,λ),maxa0µ ln(wλω + ra− a0 − wq) + ν ln(κδ)
+(1− µ− ν) ln(2− ω − η) + βV j+1(a0, 2,λ). P(2)
and
V j(a, 2,λ) = maxmaxa0µ ln(wλω + φwλω + ra− a0) + ν ln(κδ)
+(1− µ− ν) ln(2− 2ω − η) + βV j+1(a0, 2,λ),maxa0µ ln(wλω + ra− a0) + ν ln(κδ)
+(1− µ− ν) ln(2− ω − η) + βV j+1(a0, 2,λ). P(3)
Denote the female labor-force participation decision that arises from these problems
by the indicator function P j(a, τ ,λ). Here P j(a, τ ,λ) = 1 denotes the event where the
woman works. Likewise, the household’s asset decision is represented by Aj(a, τ ,λ).
The Adoption Decision: Now, suppose that a household currently does not own
the new technology. The household faces a choice about whether to adopt the new
technology in the current period or not. The decision problem facing an age-j house-
hold is
maxτ∈0,1
V j(a, τ ,λ). P(4)
14
Let T j(a,λ) represent the indicator function that summarizes the decision to adopt
the new technology or not. The solution to this problem is simple:
T j(a,λ) =
1, if V j(a, 1,λ) > V j(a, 0,λ),
0, if V j(a, 1,λ) ≤ V j(a, 0,λ).
It only applies to those agents who haven’t adopted previously. The law of motion
for technology must specify that τ j+1 = 2 if either τ j = 1 or τ j = 2.
Decision Rules: Consider generation j. Denote an age-j household’s current asset
holdings by aj and their state of technology by τ j. Now, note that for the first
generation a1 = 0. This implies that aj+1 and τ j+1 can be represented by aj+1 =
Aj(λ) and τ j+1 = Tj(λ). To see that this is the case, suppose that aj = Aj−1(λ)
and τ j = Tj−1(λ). If τ = 2 then either τ j+1 = Tj−1(λ) or τ j+1 = Tj−1(λ) + 1,
and if τ 6= 2 then τ j+1 = T j(Aj−1(λ),λ). Hence, write τ j+1 = Tj(λ). Likewise,
aj+1 = Aj(Aj−1(λ),Tj(λ),λ) ≡ Aj(λ). To start the induction, let T 1(0,λ) ≡ T1(λ)
and A1(0, T 1(λ),λ) ≡ A1(λ). Similarly, an age-j household’s participation decision
can be written asPj(λ). Now, at any point in time some households will have adopted
the new technology while others won’t have. Let λj be the threshold level of λ such
that all households of generation j with λ > λj will have adopted, while those with
λ ≤ λj won’t have.
4 Competitive Equilibrium
Market-Clearing Conditions: At each point in time all factor markets must clear.
This implies that the market demand for labor must equal the market supply of
labor. Therefore,
l = nω
ZλL(dλ) + φω
nXj=1
ZλPj(λ)L(dλ). (4)
The market supply of labor is obtained by summing males’ and females’ labor supplies
across the varying ability levels and generations. Likewise, next period’s market
15
capital stock must equal today’s purchases of assets so that
k0 =nXj=1
ZAj(λ)L(dλ). (5)
Since the market sector is competitive, factor prices are given by marginal products.
Hence,
w = (1− α)z(zl/k)−α, (6)
and
r0 = α(zl0/k0)1−α + χ. (7)
It is time to define the competitive equilibrium under study.
Definition: A stationary competitive equilibrium consists of a set of n threshold
values Λ = λ1, ...,λn, a set of allocation rules Aj(λ), Pj(λ), and Tj(λ) for
j = 1, ..., n, and a set of wage and rental rates, w and r, such that
1. The allocation rules Aj(λ) and Pj(λ) solve problem P(1) to P(3), given
w, r, and q.
2. The allocation rule Tj(λ) solves problem P(1) to P(3), given w, r, and q.
3. Factor prices clear all markets, implying that (4) to (7) hold.
Balanced Growth: Represent the pace of labor-augmenting technological progress
by γ so that γ = z0/z. Let z0 = 1 so that zt = γt. Conjecture that y, m, i, d,
and k all grow at this rate too. Also, posit that along a balanced-growth path the
aggregate stock of labor, l, is constant. This conjecture is consistent with the forms
of (1) to (3). This implies from (6) and (7) that r is constant over time, while w
grows at rate γ. It remains to be shown that Ajt+1(λ) = γAj
t(λ), Pjt+1(λ) = Pj
t(λ),
and Tjt+1(λ) = Tj
t(λ). Observe that this solution will be consistent with the factor
market-clearing conditions (4) and (5).
16
Lemma 1 Along a balanced growth path, Ajt+1(λ) = γAj
t(λ), Pjt+1(λ) = Pj
t(λ), and
Tjt+1(λ) = Tj
t(λ).
Proof. Suppose that along a balanced-growth path V j+1(γa, τ ,λ; γw) = V j+1(a, τ ,λ;w)+
[(1 − βn−j)/(1 − β)] ln γ.11 Now, by eyeballing problems P(1) to P(3) it is easy
to see that if Aj(a, τ ,λ;w) and P j(a, τ ,λ;w) are the solutions to these problems
when the state of world is (a, τ ,λ;w), then Aj(γa, τ ,λ; γw) = γAj(a, τ ,λ;w) and
P j(γa, τ ,λ; γw) = P j(a, τ ,λ;w) are the solutions when the state of the world is
given by (γa, τ ,λ; γw). It then follows that V jt (γa, τ ,λ; γw) = Vjt (a, τ ,λ;w) + [(1 −
βn−j+1)/(1− β)] ln γ. Finally, note from problem P(4) that T j(a,λ) = T j(γa,λ; γw).Therefore, if Aj(a, τ ,λ;w), P j(a, τ ,λ;w) and T j(γa,λ; γw) solve problems P(1) to
P(3) today — when the state is (a, τ ,λ;w) — then γAj(a, τ ,λ;w), P j(a, τ ,λ;w) and
T j(a,λ;w) will solve them tomorrow — when the state will be (γa, τ ,λ; γw).
5 Findings
5.1 Some Preliminary Analysis
Calibration: Take the period for the model to be 5 years. Let n = 10 so that a
household has a working life of 50 years. Clearly, ω, η, and ρ can be pinned down
from time-use data. For instance, in a week there are 112 non-sleeping hours available
per adult. If full-time work involves a 40 hour workweek, then ω = 0.36. In 1900
about 58 hours a week were spent on housework, while in 1965 roughly 15 were.
So, set η = 0.13 and ρη = 0.52. Next, pick φ = 0.60, approximately the ratio of
female to male earnings in 1980. This leaves the parameters κ, µ, and ν. In 1929
the stock of appliances was about $5,272 mil. (in 1982$) while in 1959 it was $32,882
mil. Hence, κ = 4.1. The lognormal distribution for λ was discretized so that
11Note that the household’s problem is a function of w and r. Hence, these factor prices should
be entered into the value functions. Since r is constant along a balanced-growth path it has been
suppressed in the value function.
17
λ ∈ Λ ≡ λ1, ...,λ100. The skill distribution was parameterized by setting σ = 0.70.Now, the two utility parameters µ and ν are set so the model’s balanced growth
displays the two features discussed momentarily. This required picking µ = 0.33 and
ν = 0.20.
The Household Sector, circa 1900 : Imagine that the time is 1900. Virtually no one
owns an electrical appliance. This situation can be obtained in the model by setting
the price of durables high, say q = 20 — this implies that the median person would
have to work 100 years to earn the income required to purchase modern durables.
The annual interest rate for the model is 7.1 percent. This lies above the rate of time
preference. Associated with this interest rate is a market investment to output ratio
of 0.15.
At this time in history, almost all married women stayed at home. In 1890 Goldin
(1987, Table 1) reports that only 5 percent of all married women worked. Sur-
prisingly, female labor-force participation is not a function of income when nobody
adopts the new technology. Hence, female labor-force participation must be some
fraction contained in the set 1/n, 2/n, ..., 1. With the adopted calibration, 10 per-cent (1/10×100%) of women work. The standard deviation (for the ln) of householdincome in the model is 0.70.
Lemma 2 If Tj(λ) = 0 for all j = 1, ..., n and λ ∈ Λ, then Pj(λ) = πj for all j and
λ.
Proof. Take a household of type λ. It’s market consumption decision must satisfy
the Euler equation
1
mj(λ)= βr
1
mj+1(λ), (8)
Using the household budget constraint, this implies that
mj(λ) = (βr)j−1 1− β1− βnΩ(λ),
18
where Ω(λ) is the present-value of the household’s income — at age 1 — net of the cost
of purchasing consumer durables. Since the household doesn’t adopt, Ω(λ) is given
by
Ω(λ) =nXj=1
wλω + φwλωPj(λ)
(r)j−1= wλω[
1− 1/rn1− 1/r + φ
nXj=1
Pj(λ)
(r)j−1].
It is then easy to calculate that lifetime utility is given by
V 1(0, 0,λ) = µ1− βn
1− β [ln(1− β1− βn ) + lnΩ(λ)] +
nXj=1
(j − 1)βj−1 ln(βr)+ ν 1− βn
1− β ln δ (9)
+(1− µ− ν)nXj=1
βj−1Pj(λ) ln(2− 2ω − ρη) + [1−Pj(λ)] ln(2− ω − ρη).
Now, there are 2n − 1 other possible work combinations. Let P∗j(λ) denote some
other arbitrary work profile and V∗1(0, 0,λ) represent the lifetime utility associated
with this particular participation sequence. To obtain V∗1(0, 0,λ) replace Pj(λ) with
P∗j(λ) in (9). For Pj(λ) to be optimal it must happen that V 1(0, 0,λ) ≥ V ∗1(0, 0,λ).
Observe that V 1(0, 0,λ)−V ∗1(0, 0,λ) is not a function of λ, however.12 Hence, Pj(λ)
cannot be a function of λ.
The Household Sector, circa 1980 : Now, move ahead to 1980. Almost everybody
owns electrical appliances. This situation transpires in the model when q = 0.04.
If a period is 5 years then there are about 8,800 working hours (5 yrs. × 11 mths.
× 4 wks. × 40 hrs.) per adult. Hence, the median male would only need to work
about 350 hours to purchase modern appliances. The steady-state interest rate is
7.01 percent, which lies above the rate of time preference. At this interest rate, the
investment-to-output ratio is 0.15. This is not far off from the postwar average of
0.11. Appliance investment amounts to 0.64 percent of GDP.
About one half of married women work now, in line with Goldin (1987, Table 1).
When everybody adopts the new technology immediately, female-labor participation
is a decreasing function of household income. This is proved in the lemma below.
12Note that lnΩ(λ) = ln(wλω) + ln[(1− 1/rn)/(1− 1/r) + φPnj=1 Pj(λ)/rj−1].
19
10 30 50 70 90
Type -- lowest to highest
0.5
0.6
0.7
0.8
0.9
1.0Pa
rtici
patio
n R
ate
Figure 6: Female Labor-Force Participation
Figure 6 illustrates the situation for the case when q = 0.04.Women from more well-
to-do households retire earlier. (Just multiply the participation rate by 10 to get the
period a women retires in.) The standard deviation for (the ln of) household income
is 0.71.
Lemma 3 The present value of labor-force participation,Pn
j=1 pj/rj−1, is nonin-
creasing in type, λ, holding fixed the date of adoption, m. Similarly,Pn
j=1 pj/rj−1 is
nonincreasing in m, holding fixed λ.
Proof (by contradiction). Consider two types of households, λ∗ and λ∗∗
with λ∗ < λ∗∗. Let p∗j denote the optimal participation policy associated with λ∗,
and p∗∗j represent the corresponding policy linked with λ∗∗. Analogously, let B∗1(λ)
and B∗∗1(λ) be the period-1 lefthand sides of the Bellman equations connected with
20
the policies. These can be obtained by replacing Pj(λ) in (9) with p∗j and p∗∗j,
respectively, and adding [(βm−1 − βn)/(1− β)]ν lnκ.Suppose that the hypothesis is not true. Then, there exists a λ∗ and λ∗∗ such
that B∗1(λ∗) > B∗∗1(λ∗), B∗1(λ∗∗) < B∗∗1(λ∗∗), andPn
j=1 p∗j/rj−1 <
Pnj=1 p
∗∗j/rj−1.
Now, observe from the analogue to (9) that, B∗1(λ) − B∗∗1(λ) = µ[(1 − βn)/(1 −β)]ln[Ω∗(λ)/ lnΩ∗∗(λ)]. Here Ω∗(λ) and Ω∗∗(λ) are the levels of permanent income(net of adoption cost) associated with the p∗j and p∗∗j policies. Now,
d[B∗1(λ)−B∗∗1(λ)]dλ
= µ1− βn1− β
q/rm−1φλω2Pn
j=1[p∗∗j − p∗j]/(r)j−1
Pnj=1[wλω + φwλωp
∗j/(r)j−1]− q/rm−12
Ω∗∗(λ)Ω∗(λ)
> 0.
Consequently, if B∗1(λ∗) > B∗∗1(λ∗), then B∗1(λ∗∗) > B∗∗1(λ∗∗). The desired contra-
diction obtains. The proof for the second part of the hypothesis parallels the first,
mutatis mutandis.
Pursuing Happiness: The second industrial revolution made women worse off, or
so you might believe from reading the sociology literature. The lot of families in
the artificial economy can be examined. Compare the 1900 and 1980 steady states.
As a result of new, more productive household capital, GDP rises by 24 percent.
It may be tempting to conclude that the gain in welfare must be less than this.
After all, the increase in GDP occurs because more women are working. In fact,
welfare increases by 50 percent, when measured in terms of market consumption.
The new technology leads to a 23 percent increase in market consumption, a 50
percent increase in nonmarket consumption, and a 21 percent increase in leisure.13
Now, take a family living in 1980. They will reside at some percentile in the income
13Parente, Rogerson and Wright (2000) show that when household production is incorporated
into the standard neoclassical growth model, cross-country differences in welfare are smaller than
cross-country differences in GDP. In their framework cross-country income differentials are due to
policy distortions. A tax on market activity reduces GDP. Welfare drops by less than GDP, though,
because there is an increase in nonmarket activity. In the current paper, technological progress leads
to a rise in all items in the utility function.
21
distribution and have an associated level of utility. At what spot in the 1900 income
distribution would a family have to be located in order to realize this same level of
utility? Figure 7 gives the answer. For instance, a poor family at the 20th percentile
in 1980 is as well off as someone living in the 55th percentile in 1900. Trivially, (in
a distortion-free competitive equilibrium) a family wouldn’t adopt a new technology
if it made them worse off; after all, they are pursuing happiness.14 The analysis
here models the household from the perspective that members share a common set
of preferences. Suppose they didn’t. It seems likely that any reasonable bargaining
model would predict that both males and females would share in the gain from the
second industrial revolution.
The Effect of Declining Prices: Between 1900 and 1980 the prices for household
appliances must have dropped dramatically. The time path of prices has a big impact
on adoption and participation decisions. To see this, hold the interest rate fixed at
7 percent and imagine that prices fall 5 percent a year starting from an initial value
of 2.0. Figure 8 tells the story. About 5 percent of households adopt immediately.
14This isn’t the view held by many social historians. On the one hand, they believe that “the
change from the laundry tub to the washing machine is no less profound than the change from
the hand loom to the power loom; the change from pumping water to turning on a water faucet
is no less destructive of traditional habits than the change from manual to electric calculating” —
Cowan (1976, pp. 8-9). On the other hand, they feel that a change in societal tastes, generated
by commercial interests, enslaved women into undertaking new tasks by making them feel guilty
if “their infants had not gained enough weight, embarrassed if their drains clogged, guilty if their
children went to school in soiled clothes, guilty if all the germs behind the bathroom sink are not
eradicated, guilty if they fail to notice the first sign of an oncoming cold, embarrassed if accused of
having body odor, guilty if their sons go to school without good breakfasts, guilty if their daughters
are unpopular because of old-fashioned, or unironed, or — heaven forbid — dirty dressed” – Cowan
(1976, p. 16). To most producing clean clothes for school is a utility generating activity. So, these
sociologists must believe that society has been duped by advertising campaigns, etc. into having
these tastes. Cowan (1976, p. 23) ends with “how long ... can we continue to believe that we will
have orgasms while waxing the kitchen floor.” Well according to Mark Twain, “It isn’t what we
don’t know that kills us, it’s everything we know that ain’t so.”
22
0.0 0.2 0.4 0.6 0.8 1.0
New Income Distribution -- Fractile
0.0
0.2
0.4
0.6
0.8
1.0
Old
Inco
me
Dis
tribu
tion
-- Fr
actil
e
Figure 7: A Utilitarian View of Economic Development
23
The higher a household’s type, the earlier they adopt. In order to acquire consumer
durables the woman in a household may have to go to work. That is, in line with
the lemma, adoption goes hand in hand with increased labor effort.15 In most cases
the woman goes to work before the durables are purchased. As λ rises from λ1 to
λ64 labor effort increases continuously as households adopt at successively earlier
dates. Holding fixed the adoption date, however, labor effort is decreasing in type —
as was proved in the previous lemma. For example, from λ78 to λ100 all households
adopt immediately; hence, the adoption date is fixed here. Labor effort decreases
continuously as the lemma states.
It may seem that theoretically the date of adoption should be a nonincreasing
function of λ. This is difficult to establish, though, given the lumpy nature of the
adopt and work decisions. The lumpy nature of these decisions can be partially
smoothed out by increasing the number of periods that a household lives while holding
fixed its lifespan; i.e., by shortening the length of a period.
Lemma 4 Along a balanced growth path the date of adoption is a nonincreasing
function in type, at least when the length of a period is sufficiently short.
Proof. See Appendix.
5.2 Some More Evidence
U.S. Evidence: So, what is the relationship between the adoption of appliances and
female labor-force participation? The evidence is scant. Still, here it is. Some
state-by-state data on the quantity of appliances sold is available from Electrical
Merchandising (1957). From this a crude measure of the stock of household appliances
15Take the case where the type distribution is continuous. Consider some threshold value of λ
and a local neighborhood around it. Suppose that above this value of λ the household adopts at
some date ς, while below it they adopt at some later date, say ς + j where the integer j ≥ 1. As thethreshold is crossed the adoption date jumps forward, but λ remains more or less fixed. Hence, the
lemma applies.
24
0 3 6 9
Period
0.00
0.05
0.10
0.15
0.20
Num
ber o
f Ado
pter
s in
Per
iod
0.0
0.5
1.0
1.5
Pric
e
Price
Adopters
20 50 80
Type -- lowest to highest
0.2
0.4
0.6
0.8
1.0
Labo
r-for
ce P
artic
ipat
ion
Rat
e --
fem
ales
0
2
4
6
8
Perio
d of
Ado
ptio
n
Period
Part.
Figure 8: The Effect of Prices on Adoption and Participation
25
400 600 800 1000 1200 1400 1600 1800
Stock of Appliances
0.20
0.25
0.30Fe
mal
e La
bor-F
orce
Par
ticip
atio
n
Figure 9: The State-by-State Relationship between Female Labor-Force Participation
and Appliance Ownership
per family can be constructed for each state. In particular, this source presents data
for washers, dryers, refrigerators, electric stoves, freezers, ironers, and electric water
heaters. For each appliance the total numbers of units sold over the twelve year
period 1946-1957 is reported. The resulting numbers can then be summed across
appliances, after multiplying each figure by the price of the appliance. These figures
can be normalized by the number of families in the state, as reported in the 1950
U.S. Decennial Census, to obtain a measure of household capital per family. State-
by-state female labor-force participation numbers can be computed from the 1950
census. The relationship between these two series is plotted in Figure 9. There is
no question that, visually, appliance ownership is positively associated with female
labor-force participation.
To test the robustness of this relationship, female labor-force participation (part)
26
will be regressed against appliance ownership (appl), plus some additional control
variables. The control variables are the ratio of females with some high school educa-
tion in the state (edu), per-capita income in the state (inc), the extent of urbanization
(urb), and some regional dummies.16 The result obtained is
part = −0.01(0.00032)
+0.000040(0.000023)
× appl+ 0.365869(0.221326)
× edu+ 0.043235(0.071375)
× log(inc)
+ 0.000629(0.000344)
× urb+ dummies,
with r2 = 0.60, σ = 0.024, obs = 48, d.f. = 40.
All coefficients have the expected signs. All are significant, except for the level of per-
capita income in a state. As can be seen, appliance ownership is positively associated
with female labor-force participation. The coefficient, if interpreted literally, implies
that a $1,000 increase in the per-capital stock of appliances in a state will be associated
with a 4.0 percentage point rise in female labor-force participation.
International Evidence: Countries where new durable goods are expensive tend
to have low levels of female labor-force participation. Here is the evidence. The Penn
World Table (Version 5.6) presents some national income account statistics for a
number of countries. By dividing the price of investment goods through by the GDP
deflator, a measure of the relative price of durables can be obtained. Strictly speaking
one would like a measure of the price of household equipment. This isn’t available.
The price of new equipment in the business sector is probably a reasonable proxy
for the price of new equipment in the household sector. The behavior of the relative
price of automobiles, computers, refrigerators, stoves, etc. used in the business sector
across time and space is likely to be similar to the relative price of these goods
used in the household sector. A measure of female labor-force participation can be
computed using data available from the Economically Active Population, 1950-2010,
a publication of the International Labor Office. In particular, the ratio of female
employees to total employees in industry and services will be taken as proxy for16Regions are classified in line with Barro and Sala-i-Martin (1995, Table 10.4)
27
female labor-force participation. This excludes the agricultural sector. Handling the
agricultural sector is a tricky issue. First, the ILO data treats any woman who works
more than one hour a week in agriculture as being employed there. For less-developed
countries this weak restriction will be satisfied by most rural women. The tendency
to count rural adults as agricultural workers, irrespective of the time they allocate
to this sector, has been noted by Parente, Rogerson and Wright (2000). Second, in
less-developed countries a lot of agriculture is really household production, at least
for the purposes here. A panel data set of 128 countries can be complied from these
two data sources. The time interval is a decade, starting with 1950 and ending in
1990. For some countries an observation was not available for each decade.
To judge the significance of the relationship, female labor-force participation
(part) can be regressed against the relative price of durables (price) and some control
variables. The control variables are GDP per worker (inc) and dummy variables for
decades and regions.17 The result is:
part = 0.27− 0.011(0.0036)
× price+ 0.022(0.0121)
× inc+ dummies,
r2 = 0.46, σ = 0.08, obs = 512, d.f. = 499.
The price and income variables take the expected sign. Both are significant18.
The cross-sectional relationship between durable goods prices and female labor-
force participation for 1990 in plotted Figure 10. Here, the relative price of durables
is deflated by GDP per worker to obtain a measure of the time price of durables.
It’s easy to see that higher durable goods prices are associated with a lower level of
female labor-force participation. The figure also graphs the steady-state relationship
between household appliance prices and female labor-force participation predicted by
17The seven regions are Africa (except North Africa), Asia (sans the Middle East), Europe, Middle
East (plus North Africa), North and Central America, Pacific Region, and South America. This
classification is taken from Barro and Sala-i-Martin (1995, Table 10.3).18For 1990 there is data on the fraction of the population with a primary or secondary education.
These variables turned out to be insignificant.
28
-4.5 -4.0 -3.5 -3.0 -2.5 -2.0 -1.5
Ln(q)
0.1
0.2
0.3
0.4
0.5
Fem
ale
Labo
r-Fo
rce
Par
ticip
atio
n
Model
Figure 10: The World-Wide Relationship between the Price of New Durables and
Female Labor-Force Participation, 1990
the model.19 The relationship generated by the model is not dissimilar to the one
found in the international data. This exercise heroically assumes that each country
was resting in its steady-state in the last decade. Don’t take this seriously. The
mapping generated by the model is shown just to illustrate how the model can be
used to shed light on some historical and geographical facts.
5.3 The Durable Goods Revolution
The Computational Experiment : The time is 1900. The age of electricity has just
dawned. This era herds in many new household goods: the dryers, frozen foods, hot
water, refrigerators, washing machines, etc. What will be the effect in the artificial
19The price series inputted into the model has been normalized (multiplied by scalar) to make the
price for U.S. in 1990 equal to 0.04.
29
economy? To answer this, the transition path from the 1900 steady state to the 1980
one will be analyzed. This experiment is in the spirit of King and Rebelo’s (1993)
analysis of the inability of the standard neoclassical growth model to explain the rise
of postwar Japan, Hansen and Prescott’s (1999) study of the pickup in growth from
1800 on, and Caselli and Coleman’s (forth.) research on the catch up of the southern
states.
To do this, a time path for durable goods prices must be inputted into the model.
Hard numbers are hard to come by, but Figure 11 plots several price series for appli-
ances. The NIPA measure for appliances declines at about 2.2 percent a year, relative
to the GDP deflator. This is likely to underestimate the decline in prices because
it doesn’t control for quality improvement in goods. The figure also shows Gordon’s
(1990, Table 7.23) quality-adjusted price index for eight appliances, viz refrigerators,
air conditioners, washing machines, clothes dryers, TV sets, dishwashers, microwaves,
and VCR’s. This series drops at 8.5 percent a year versus only 3.5 percent for the
standard PPI measure. Note that time prices will decline at an even faster clip, since
wages have risen over time. Quality-adjusted time prices for some select appliances
are shown in Figure 12.20 Assume, then, that time prices decline on average at about
5 percent a year for the first 100 years. The analysis also presumes that agents have
perfect foresight; a heroic assumption, for sure.
Adoption and Labor-Force Participation: The new appliances catch on slowly at
first. This can be seen from Figure 13, which plots the diffusion curve. It takes 50
years for half of the population to own the durables. Female labor-force participation
rises along with adoption. This is shown in Figure 13, too. Only wealthy households
— high types — can afford to buy when prices are high. The diffusion curves for three
age-averaged types of households, λ1, λ50, and λ100, are graphed in Figure 14.
Pursuing Happiness, Again: The increase in GDP due to the durable goods rev-
olution is shown in Figure 15. This rise occurs solely because of the increase in
20These are based on series contained in Gordon (1990, Tables 7.4, 7.12, 7.15, and 7.22)
30
1920 1930 1940 1950 1960 1970 1980 1990Year
0.1
0.5
0.9
1.3
Pric
e of
App
lianc
es --
NIP
A
0.0
0.2
0.4
0.6
0.8
1.0
Pric
e of
App
lianc
es --
Gor
don
and
PPI
NIPA, 2.2%
Gordon, 8.5%
PPI, 3.5%
Figure 11: Decline in the Relative Price of Appliances
31
1950 1960 1970 1980
Year
0.0
0.2
0.4
0.6
0.8
1.0
Tim
e Pr
ice
of A
pplia
nces
, q
Refridgerators, 8.1%
Microwaves, 12.8%
Dryers, 8.7%
Washers, 5.7%
Dishwashers, 5.7%
Figure 12: Prices for Some Select Appliances
32
0 40 80 120
Time, years
0
20
40
60
80
100
Perc
enta
ge
0
2
4
6
8
10
Pric
e, q
Diffusion
Participation
Price
Figure 13: Transitional Dynamics — Diffusion and Female Labor-Force Participation
33
0 20 40 60 80 100
Time, years
0
20
40
60
80
100
Perc
enta
ge
Median
Lowest
Highest
Figure 14: Diffusion by Type of Household
34
0 40 80 120
Time, years
2.5
2.6
2.7
2.8
2.9
3.0
Gdp
0.0
0.1
0.2
0.3
0.4
0.5
Com
pens
atin
g Va
riatio
n
c.v.
gdp
Figure 15: The Gain in GDP and Welfare
35
female labor-force participation. Other than the durable goods revolution there is no
technological progress. The associated rise in welfare — for the flow of new households
into the economy — is also plotted in this figure. Again, the gain in welfare is greater
than the increase in GDP.
Between 1929 and 1999 per-capita real GDP grew at 2.2 percent per annum. Take
this as reflective of the average rate of growth over the last century. In the model
over a 50 year period GDP grows by about 0.35 percent per annum, while for an 80
year period it grows by about 0.22 percent. Therefore, according to the model, the
durable goods revolution can be thought of as accounting for about 15 percent of
growth, say, between 1920 and 1970, or about 10 percent of growth between 1900 to
1980.
6 Conclusions
“For ages woman was man’s chattel, and in such condition progress for
her was impossible; now she is emerging into real sex independence, and
the resulting outlook is a dazzling one. This must be credited very largely
to progression in mechanics; more especially to progression in electrical
mechanics.
Under these new influences woman’s brain will change and achieve
new capabilities, both of effort and accomplishment.”
Thomas Alva Edison, as interviewed in Good Housekeeping Magazine,
LV, no. 4 (October 1912, p. 440)
Did technological progress unlock the manacles chaining women to the home?
That’s the question posed here. Some may argue that the increase in female labor-
force participation was due to a change in social norms, say spawned by the women’s
liberation movement. After reviewing public opinion poll evidence, Oppenheimer
(1970, p. 51) concludes “it seems unlikely that we can attribute much of the enor-
36
mous postwar increases in married women’s labor force participation to a change in
attitudes about the propriety of their working.”21 Besides without the labor-saving
household capital ushered in by the second industrial revolution it simply would not
have been feasible for many women to spend more time outside of the home, notwith-
standing any shift in societal attitudes. While sociology may have acted as a fertilizer,
the seed of women’s liberation came from economics.
A Appendix
Growth Transformation: Consider the consumption decision for an age-1 household.
It must satisfy the Euler equation
1
mj= βr
1
mj+1, (10)
wheremj is the household’s consumption at age j. The household’s budget constraint
will read
m1 +m2
r+m3
r2+ ...+
mn
rn−1= Ω,
where Ω is the household’s permanent income, net of the cost of purchasing the
consumer durables. The Euler equation (10) implies that mj+1 = βrmj = (βr)jm1.
Therefore,
m1 =1− β1− βnΩ.
21Unfortunately the questions asked are different both across polling organizations and years —
see Oppenheimer (1970, Table 2.10). In 1960 only 34 percent of respondents answered approvingly
to the following question: “There are many wives who have jobs these days. Do you think it is
a good thing for a wife to work or a bad thing, or what? Why do you say so?” A poll in 1937
asked the question “Do you approve a married women earning money in business or industry if
she has a husband capable of supporting her?” Eighteen percent of respondents approved. The
same percentage answered favorably to a similar question in 1945. When questions were qualified
to indicate some sort of financial need — support for children, a new marriage, etc. — the percentage
of favorable responses went up.
37
Adding growth would not seem to change this equation much. All variables that
grow along a balanced path should be transformed to obtain a stationary representa-
tion. Define ajt = ajt/γ
t, bmjt = m
jt/γ
t, bwt = wt/γt, and bΩt = Ωt/γt. Then, the Eulerequation would appear as
1bmjt
= β(r/γ)1bmj+1t+1
.
The household’s budget constraint now reads
bm1t +
bm2t+1
(r/γ)+
bm3t+2
(r/γ)2+ ...+
bmnt+n−1
(r/γ)n−1= bΩt.
Therefore,
bm1 =1− β1− βn
bΩ.Here
bΩt = nXj=1
bwλω + φbwλωP j(λ)− bwqT j(λ)(r/γ)j−1
,
where
bw = (1− α)[r/γ − χα
]α/(α−1), (11)
and
r/γ = α(l/bk)1−α + χ.Last, the market clearing condition for capital would appear as
bk0 = nXj=1
Z bAj(λ)L(dλ).
Transitional Dynamics: Imagine that the economy is resting in some initial steady-
state. Since the electric age hasn’t emerged yet, all households are using primitive
durables at home. Now, suppose that suddenly new household durables are invented.
38
At the time of the durable goods revolution, the initial state of the economy is de-
scribed by s = (a2(λ), a3(λ), ..., an(λ)). The system will eventually converge to a
new steady state represented by s = (a2∗(λ), a3∗(λ), ..., an∗(λ)), where an asterisk
attached to a variable signifies its value in the new steady state. Assume that this
convergence will take place within e + 1 periods. The time path of prices for these
goods is given by q1 > q2 > ... ≥ qe−m = qe+1 = q∗. The algorithm used to compute
the model’s transitional dynamics can now be outlined.
1. Enter each iteration i with a guess for the interest rate path, or −→r 2 = rte+1t=2 .
Denote this guess by −→r i2 = rite+1t=2 . Using (11) this will imply a guess for wages
−→w i2 = wite+1
t=2 . Note that by assumption re+1 = r∗ and we+1 = w
∗.
2. Using this guess, solve out for−→s 1 = ste+1t=1 . This is done as in the manner
below:
(a) Enter period t with state of world st, which was computed in the previous
period t − 1. For each j and λ solve the household’s decision problemsto obtain Aj
t(λ), Pjt(λ), and Tj
t(λ). Set st+1 = (A2t (λ), A
3t (λ), ..., A
nt (λ)).
Move onto period t+ 1 (unless t = e, in which case you’re finished).
(b) For age-j agent, with skill level λ, permanent income in period t will be
given by the formula22
bΩjt(λ) = ritaj + n−jXm=0
bwt+m[λω + φλωP j+m(λ)− qt+mT j+m(λ)]Πt+mk=t+1(r
ik/γ)
.
(c) The period−t market clearing wage can be obtained by finding the bwt suchthat (4) holds. Set bwt+m = wit+m for m > 0.
Compute a revised guess for the interest rate path −→r 2, denoted by −→r i+12 , using
the formula
ri+1t+1/γ = α(lt+1/bkt+1)
1−α + χ.
22In this formula, Πtn=t+1 ≡ 1.
39
It may be better to set
ri+1t+1/γ = ϑ[α(lt+1/bkt+1)
1−α + χ] + (1− ϑ)rit+1/γ, for 0 < ϑ < 1.
Lemma 4: Along a balanced growth path the date of adoption is a nonincreasing
function in type, at least when the length of a period is sufficiently short.
Proof. Consider the continuous-time analogue to the adopt/work problem framed
by P(1) to P(4). Let the date of adoption chosen by the household be represented by
α. The household will choose an interval [σ, ε] ⊆ [0, n] over which to work. Here σdenotes the start date for working and ε denotes the end date. As an example of how
things work, take the case where σ = 0 < α < ε < n. Here the woman in a household
starts working immediately, builds up some resources to purchase durables at age α,
and then retires at ε. A type-λ household’s decision problem is
maxα,εµ1− e
−βn
β[ln(
β
1− e−βn ) + lnΩ(λ)] + µZ n
0
(r − β)je−βjdj
+ν1− e−βn
βln δ + ν lnκ
Z n
α
e−βjdj + (1− µ− ν)[ln(2− 2ω − ρη)Z α
0
e−βjdj
+ ln(2− 2ω − η)Z ε
α
e−βjdj + ln(2− ω − η)Z n
ε
e−βjdj],
subject to
Ω(λ) = wλω1− e−rn
r+ φwλω
Z ε
0
e−rjdj − qwe−ια.
Now, r represents the net interest rate and β is the rate of time preference.
The first-order conditions to this problem are:
µ1− e−βnβ
wqιe−(r−β)α = [(1− µ− ν) ln( 2− 2ω − η2− 2ω − ρη ) + ν lnκ]Ω(λ),
and
µ1− e−βn
βφwλωe−(r−β)ε = (1− µ− ν) ln( 2− ω − η
2− 2ω − η )Ω(λ).
40
Undertaking the requisite comparative statics exercise gives
dα
dλ= [(1− µ− ν) ln( 2− 2ω − η
2− 2ω − ρη ) + ν lnκ](wω/r)[1− e−rn + φ(1− e−εr)]
× (r − β)µ(1− e−rn)φwλωe−(r−β)ε
β det(H)< 0,
where H is the 2 × 2 Hessian associated with the maximum problem. To sign the
above expression, note that the second-order conditions for a maximum necessitate
that the matrix H is negative semidefinite. Necessary conditions for this to transpire
are that det(H) ≥ 0 and H11,H22 ≤ 0, where H11 and H22 are the entries in upper
left and lower righthand corner of H. When r > β it is easy to see that dα/dλ < 0.
When r < β it can be shown that a maximum cannot obtain, since H11,H22 < 0 then
imply det(H) < 0. There are many other cases to consider, but they all proceed in
the same manner. (Basically, the rest of the proof is a boring taxonomy.)
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